Question

Consider the loop transfer function \(\frac {K(s+6)}{(s+3)(s+5)}\). In the root locus diagram the centroid will be located at:

• A. -4
• B. -1
• C. -2
• D. -3

Hint:

Root Locus Centroid. The centroid \(\sigma_A\) is the intersection point of the asymptotes on the real axis. Formula: \(\sigma_A = (\sum Poles - \sum Zeros) / (n - m)\), where n is the number of poles and m is the number of zeros.

Answer 1

The Correct Option is C

The root locus diagram shows how the poles of the closed-loop system change as the gain K varies
The asymptotes of the root locus indicate the behavior of the poles for large K
These asymptotes emanate from a point on the real axis called the centroid (\(\sigma_A\))
The loop transfer function is given by \(G(s)H(s) = \frac{K(s+6)}{(s+3)(s+5)}\)
Identify the poles and zeros of the loop transfer function: - Zeros (roots of numerator): \(z_1 = -6\)
Number of zeros \(m = 1\)
- Poles (roots of denominator): \(p_1 = -3\), \(p_2 = -5\)
Number of poles \(n = 2\)
The centroid is calculated using the formula: $$ \sigma_A = \frac{\sum (\text{Real part of poles}) - \sum (\text{Real part of zeros})}{n - m} $$ $$ \sigma_A = \frac{(p_1 + p_2) - (z_1)}{n - m} $$ $$ \sigma_A = \frac{(-3 + (-5)) - (-6)}{2 - 1} $$ $$ \sigma_A = \frac{(-8) - (-6)}{1} = \frac{-8 + 6}{1} = -2 $$ The centroid is located at -2 on the real axis